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Introduction to Management Science
8th Edition
by
Bernard W. Taylor III
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Chapter 5
Integer Programming
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Chapter Topics
Integer Programming (IP) Models
Integer Programming Graphical Solution
Computer Solution of Integer Programming Problems With Excel and QM for Windows
Chapter 5 - Integer Programming
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Integer Programming Models
Types of Models
Total Integer Model: All decision variables required to have integer solution values.
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0-1 Integer Model: All decision variables required to have integer values of zero or one.
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Mixed Integer Model: Some of the decision variables (but not all) required to have integer values.
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Chapter 5 - Integer Programming
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A Total Integer Model (1 of 2)
Machine shop obtaining new presses and lathes.
Marginal profitability: each press $100/day; each lathe $150/day.
Resource constraints: $40,000, 200 sq. ft. floor space.
Machine purchase prices and space requirements:
Chapter 5 - Integer Programming
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A Total Integer Model (2 of 2)
Integer Programming Model:
Maximize Z = $100x1 + $150x2
subject to:
8,000x1 + 4,000x2 $40,000
15x1 + 30x2 200 ft2
x1, x2 0 and integer
x1 = number of presses
x2 = number of lathes
Chapter 5 - Integer Programming
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A 0 - 1 Integer Model (1 of 2)
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Data:
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Selection constraint: either swimming pool or tennis center (not both).
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Resource constraints: $120,000 budget; 12 acres of land.
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Recreation facilities selection to maximize daily usage by residents.
Chapter 5 - Integer Programming
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Integer Programming Model:
Maximize Z = 300x1 + 90x2 + 400x3 + 150x
subject to:
$35,000x1 + 10,000x2 + 25,000x3 + 90,000x4 $120,000
4x1 + 2x2 + 7x3 + 3x3 12 acres
x1 + x2 1 facility
x1, x2, x3, x4 = 0 or 1
x1 = construction of a swimming pool
x2 = construction of a tennis center
x3 = construction of an athletic field
x4 = construction of a gymnasium
A 0 - 1 Integer Model (2 of 2)
Chapter 5 - Integer Programming
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A Mixed Integer Model (1 of 2)
$250,000 available for investments providing greatest return after one year.
Data:
Condominium cost $50,000/unit, $9,000 profit if sold after one year.
Land cost $12,000/ acre, $1,500 profit if sold after one year.
Municipal bond cost $8,000/bond, $1,000 profit if sold after one year.
Only 4 condominiums, 15 acres of land, and 20 municipal bonds available.
Chapter 5 - Integer Programming
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Integer Programming Model:
Maximize Z = $9,000x1 + 1,500x2 + 1,000x3
subject to:
50,000x1 + 12,000x2 + 8,000x3 $250,000
x1 4 condominiums
x2 15 acres
x3 20 bonds
x2 0
x1, x3 0 and integer
x1 = condominiums purchased
x2 = acres of land purchased
x3 = bonds purchased
A Mixed Integer Model (2 of 2)
Chapter 5 - Integer Programming
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Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution
A feasible solution is ensured by rounding down non-integer solution values but may result in a less than optimal (sub-optimal) solution.
Integer Programming Graphical Solution
Chapter 5 - Integer Programming